Question:
If \[ a+b+c=0 \] find:
\[ \frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \]
Solution:
Taking LCM:
\[ \frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} = \frac{a^3+b^3+c^3}{abc} \]
Using identity:
\[ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]
Since \[ a+b+c=0 \]
\[ a^3+b^3+c^3-3abc=0 \]
\[ a^3+b^3+c^3=3abc \]
Substituting:
\[ \frac{a^3+b^3+c^3}{abc} = \frac{3abc}{abc} \]
\[ =3 \]